Approximating the Bipartite TSP and its Biased Generalization Preprint? Aleksandar Shurbevski1 , Hiroshi Nagamochi1 , and Yoshiyuki Karuno2 1

2

Department of Applied Mathematics and Physics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan {shurbevski,nag}@amp.i.kyoto-u.ac.jp Department of Mechanical and System Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan [email protected]

Abstract. We examine a generalization of the symmetric bipartite traveling salesman problem (TSP) with quadrangle inequality, by extending the cost function of a Hamiltonian tour to include a bias factor β ≥ 1. The bias factor is known and given as a part of the input. We propose a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs, and show that it is an approximation algorithm for the gener1+λ alized problem with an approximation ratio of 1 + β+λ , where λ is a real parameter dependent on the problem instance. This expression is bounded above by a constant 2, for any positive real λ and β ≥ 1, which improves a previously reported approximation ratio of 16/7. As a part of a composite heuristic, the proposed procedure can contribute to an 2 , where ζ is an approximation ratio approximation ratio of 1 + ζ+β(2−ζ) for the metric TSP.

Keywords: combinatorial optimization; approximation algorithm; matroid intersection; material handling robot; bipartite TSP; biased cost

1

Introduction

The traveling salesman problem (TSP) is a landmark problem in combinatorial optimization (e.g., Cook [7]). Its bipartite analogue is as follows. Given a bipartite graph G = (B, W ; E) with an edge weight function w : E → R+ , find a shortest (w.r.t. w) alternating tour which visits every point of B ∪ W exactly once. We assume that the weight function w is symmetric and satisfies the quadrangle inequality (the bipartite analogue of the triangle inequality, see Eqs. (6) and (7)). We do so not only because do the above conditions suffice in many cases based on real world scenarios, but also because just like the TSP, it is hopeless ?

Full paper DOI:10.1007/978-3-319-04657-0_8, available on line at http://link. springer.com/chapter/10.1007%2F978-3-319-04657-0_8

2

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

to approximate the bipartite TSP within a constant factor in the general case, assuming that P 6= NP [10, 15]. The bipartite TSP has justly attracted attention due to its applicability in typical industrial settings where pick and place or grasp and delivery robots are employed with some material handling tasks [3–5, 10, 11, 18]. For the symmetric case, the best known approximation factor 2 has been independently reported by Chalasani et al. [5] and Frank et al. [10]. With a specific industrial scenario in mind, the bipartite TSP has been extended to account for additional transportation effort [17]. The motivation behind this generalization is to assign certain “difficulty” when transporting an item versus simply moving through space. This has been achieved by the means of a bias factor β ≥ 1. The bias factor extends the weight function w as follows ( βw(u, v), u ∈ B, v ∈ W, w(u, e v) = (1) w(u, v), u ∈ W, v ∈ B. To the best of our knowledge, Shurbevski et al. [17] gave the first account examining the presence of a bias factor, and at the same time, demonstrated a constant 16/7-factor approximation algorithm. The previously reported approximation ratio of 16/7 has been achieved by a composite heuristic (see, e.g., [14] for terminology relating to composite heuristics). In this paper, we present a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs and show that for the biased case it is an approximation algorithm with an approximation ratio of 1+λ (2) 1+ β+λ where λ is a real parameter which depends on the problem instance and cannot be known upfront. On one hand, the above expression is bounded by a constant 2 for any positive real λ and β ≥ 1, thus the proposed algorithm has a constant factor approximation ratio, improving the one from [17]. On the other hand, for a finite λ, the above expression approaches 1 as β grows larger. The presented approach by itself does not rely on approximating the metric TSP, however it can be used as a part of a composite heuristic to achieve an approximation ratio of 2 1+ , (3) ζ + β(2 − ζ) where 1 < ζ ≤ 2 is an approximation ratio for the metric TSP. The expression from Eq. (3) is also bounded above by a constant 2, but it is not dependent on an instance-specific parameter, and has a clear relationship with the bias β for a fixed ζ < 2.

2

Preliminaries

The set of reals (resp., nonnegative reals) is denoted by R (resp., R+ ).

Approximating the Bipartite TSP and its Biased Generalization

3

In general, for a minimization problem P, let P ∗ be the value of an optimal solution. An approximation algorithm ALG is such that for any instance of P, it can produce a feasible solution of value P 0 . We call the value 0 P (4) αALG = sup P∗ the approximation factor of algorithm ALG, and usually say that ALG is an αALG -approximation algorithm. We use standard notation from graph theory; the ordered pair G = (V, E) is a connected undirected graph. The vertex set and the edge set of G are denoted by V (G) and E(G), respectively. We allow for parallel edges, or think of G = (V, E) as a multigraph. Thus, E(G) is a multiset of elements in V ×V . (We will make use of the multiset Uk sum function, denoted by the symbol ], as well as the shorthand k · E for i=1 E.) We use {u, v}, u, v ∈ V (G) to reference any and all e ∈ E(G) such that e is incident with u and v. For u ∈ V (G), dG (u) denotes the degree of the node u in the graph G. A graph is weighted if we are given some weight function w : E(G) of edges E 0 ⊆ E, P → R+ over the graph’s edges. For any subset 0 0 w(E for a subgraph G of G, w(G0 ) denotes P ) denotes e∈E 0 w(e). Similarly, 0 0 e∈E(G0 ) w(e). A subgraph G of G is spanning if V (G ) = V (G). We assume that all parallel edges are of the same weight, and ∀e ∈ E(G), e = {u, v}, we equate the expressions w(e) and w(u, v). The weight function w is said to be symmetric if w(u, v) = w(v, u), ∀e = {u, v} ∈ E(G), (5) and that it satisfies the triangle inequality if w(u, v) ≤ w(u, q) + w(q, v),

∀q, u, v ∈ V (G).

(6)

A complete bipartite graph G = (B, W ; E) is such that V (G) = B ∪ W , B ∩ W = ∅, and E(G) = B × W . A property similar to the triangle inequality can be extended over complete bipartite graphs, into the quadrangle inequality w(u, v) ≤ w(u, q) + w(q, y) + w(y, v),

∀u, y ∈ B, q, v ∈ W.

(7)

For a complete graph induced by a set of vertices B, we write G[B]. By definition, V (G[B]) = B and E(G[B]) = B × B. Let G = (B, W ; E) be a given bipartite graph with an edge weight function w : E(G) → R+ , and G[B] is exactly the complete graph induced by the partition B. Often in practice the vertex sets are in fact points from some metric space and the distance in this space serves as an edge weight function. In such a case, the edge weight function of G[B] is defined by the distance function in the metric space. However, if we are only given a bipartite graph G = (B, W ; E) with an edge weight function w : E(G) → R+ , we can extend the edge weight function over the induced graph G[B] as such w(u, y) = min {w(u, q) + w(q, y)} q∈W

∀ u, y ∈ B.

(8)

4

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Lemma 1. For a given complete bipartite graph G(B, W ; E) with a symmetric edge weight function w : E(G) → R+ satisfying the quadrangle inequality, let G[B] be the complete graph induced by the vertex partition B. The extension of w as an edge weight function of G[B] of Eq. (8) is symmetric and satisfies the triangle inequality. Given a graph G = (V, E), a Hamiltonian cycle H is a connected spanning subgraph of G such that dH (u) = 2,

∀u ∈ V (G).

(9)

The problem of finding a Hamiltonian cycle H of minimum w(H) is commonly referred to as the traveling salesman problem (TSP). For a complete bipartite graph G = (B, W ; E), with |B| = |W |, let n := |B|(= |W |) and let σ and τ be permutations on the points of B and W , respectively. A traversal of a Hamiltonian cycle H in G is of the form σ(1) → τ (1) → σ(2) → · · · → τ (n − 1) → σ(n) → τ (n) → σ(1).

(10)

We term Hamiltonian cycles in bipartite graphs alternating, for points in B and W appear alternately. When using an indexing device i = 1, . . . , n, we allow it to wrap around, i.e. ( i + n, i ≤ 0, i := i − n, i > n. As subgraphs of G, Hamiltonian cycles are undirected. However, once we settle for a way to traverse them, they assume an orientation. In addition to the edge weight w, we are concerned with a bias factor β ≥ 1. The bias factor impacts bipartite graphs as in Eq. (1). Assuming a traversal orientation as in Eq. (10), we introduce the biased cost L for alternating cycles L(H) = β

n X i=1

w(σ(i), τ (i)) +

n X

w(τ (i), σ(i + 1)).

(11)

i=1

We are now prepared to state the bipartite analogue of the metric TSP in face of the bias factor β ≥ 1. The biased bipartite traveling salesman problem – BBTSP Instance: A complete bipartite graph G = (B, W ; E), with |B| = |W |, a symmetric weight function w : E(G) → R+ which satisfies the quadrangle inequality, and a bias factor β ≥ 1. Task: Find an alternating Hamiltonian cycle H ∗ in G such that L(H ∗ ) is minimized. In this paper we focus exclusively on the version of the BBTSP where the edge weight function w is symmetric and satisfies the quadrangle inequality. We settle for this limitation because it has been shown [1, 10, 13, 15] that the bipartite TSP is not only NP-hard to solve, but also that in the general case, there is no constant factor approximation under the assumption that P 6= NP.

Approximating the Bipartite TSP and its Biased Generalization

3

5

Building Blocks

In this section we will exhibit some of the known lower bounds on the value of an optimal solution for the BBTSP, as well as add a few new insights into their correlations. The presented lower bounds are structures well known in combinatorial optimization, and will serve as building blocks for a new procedure for constructing alternating Hamiltonian cycles in bipartite graphs. 3.1

Known Lower Bounds of the BBTSP

We present some of the observations made in [17] concerning the lower bounds of an optimal solution for the BBTSP. Our analysis mainly concerns two combinatorial structures in bipartite graphs; perfect matchings, and alternating spanning trees. We will just briefly state their definitions. Let G = (B, W ; E) be a (weighted) complete bipartite graph with an edge weight function w : E(G) → R+ and |B| = |W | =: n. The edge weight function w is assumed symmetric and satisfying the quadrangle inequality (Eq. (7)). A perfect matching M ⊂ E(G) is such that there is exactly one edge in M incident with any u ∈ V (G). An alternating spanning tree T (illustrated in Fig. 1(a)) is a connected acyclic spanning subgraph of G such that dT (u) ≤ 2, ∀u ∈ B. (12) Both perfect matchings and alternating spanning trees are well studied combinatorial structures, e.g., [12, 16], and there exist polynomial time algorithms for computing perfect matchings and alternating spanning trees (of minimum weight) in bipartite graphs. Henceforth, let M ∗ denote a perfect matching in G of minimum weight w(M ∗ ), and T ∗ an alternating spanning tree with minimum w(T ∗ ). Given an instance of the BBTSP, let H ∗ be an optimal solution, which minimizes the biased cost L(H ∗ ). The edges of E(H ∗ ) can be decomposed into two −→ ←− disjoint perfect matchings, H ∗ and H ∗ , as in Fig. 1(b). Without loss of general−→ ity, we assume H ∗ is to be traversed as indicated by arrows in Fig. 1(b), and H ∗ solely accounts for the bias term. The biased path cost L(H ∗ ) is given by

It surely holds

−→ ←− L(H ∗ ) = βw(H ∗ ) + w(H ∗ ).

(13)

−→ ←− w(M ∗ ) ≤ w(H ∗ ) ≤ w(H ∗ ).

(14)

Concerning alternating spanning trees in G, note that w(T ∗ ) is a lower bound of the weight of an alternating Hamiltonian cycle disregarding the bias factor, i.e., −→ ←− w(T ∗ ) ≤ w(H ∗ ) + w(H ∗ ).

(15)

6

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

v :B :W

→

H*

Ĉ

←* H

(a) (b) Fig. 1. (a) An alternating spanning tree T . (b) A minimum cost alternating Hamil−→ ←− tonian path H ∗ of G. The subsets of edges H ∗ (bold gray arrows) and H ∗ (slender ˆ on G[B] is given in black arrows) form two disjoint perfect matchings. The shortcut C dashed lines.

Observing the graph G[B] induced by the vertex partition B, we can see that an alternating Hamiltonian cycle in G does in fact visit each vertex in B exactly once, and can be shortcut to a Hamiltonian cycle of G[B]. We will use the extended w from Eq. (8) for G[B]. For an optimal alternating Hamiltonian cycle H ∗ , let Cˆ be the resulting shortcut, as given in Fig. 1(b). Due to Eq. (8) we have → ←− ˆ ≤ w(− w(C) H ∗ ) + w(H ∗ ). (16) Consequently, for an optimal (w.r.t. the extended w) Hamiltonian cycle C ∗ in G[B] it holds → ←− ˆ ≤ w(− w(C ∗ ) ≤ w(C) H ∗ ) + w(H ∗ ). (17) 3.2

Further Observations

We would like to bring a special attention to an observation with respect to the structures presented above, alternating spanning trees and perfect matchings. Let M ∗ and T ∗ be a minimum weight perfect matching and a minimum weight alternating spanning tree in a given bipartite graph G, respectively. Owing to its special structure any alternating spanning tree in G contains a perfect matching. Therefore, let T M ⊂ E(T ∗ ) denote the edge set forming a perfect matching, and T > the remaining edges of the alternating tree, i.e. T > = E(T ∗ )\T M . It simply holds w(T ∗ ) = w(T M ) + w(T > ). (18) We present our view of the structure of an optimal solution, H ∗ , with L(H ∗ ) = −→ ←− βw(H ∗ ) + w(H ∗ ), (see Eq. (13)). We introduce a parameter λ ∈ R+ as ←− w(H ∗ ) λ= (19) −→ . w(H ∗ ) Then, for the cost of an optimal tour H ∗ we can write −→ L(H ∗ ) = (β + λ)w(H ∗ ).

(20)

Approximating the Bipartite TSP and its Biased Generalization

7

For a given instance of the BBTSP, the value of the parameter λ cannot be known without solving it exactly. However, for the purpose of our exposition, it suffices that λ ∈ R+ .

4

A New Approximation Algorithm

In this section we present a procedure for building an alternating Hamiltonian cycle in a given bipartite graph G = (B, W ; E) with |B| = |W |. We show that if the graph G is endowed with a positive symmetric edge weight function w which satisfies the quadrangle inequality, this procedure can be used as an approximation algorithm for the BBTSP. The procedure for building an alternating Hamiltonian cycle does not rely on approximating the metric TSP. 4.1

Construction

Let G = (B, W ; E), be a bipartite graph with |B| = |W | =: n. Let w : E(G) → R+ be a symmetric edge weight function satisfying the quadrangle inequality. Let M ∗ and T ∗ be a perfect matching and an alternating spanning tree in G of minimum w(M ∗ ) and w(T ∗ ), respectively. We bring to attention the union of M ∗ and T ∗ . As observed in Section 3.1, the alternating tree T ∗ contains a perfect matching, T M . The union of T M and M ∗ forms a cycle cover of G. Let there be k ≤ n individual cycles, which we will denote by R := {Ri : i = 1, 2, . . . , k}. We can think of elements of R as nodes, and define a graph GR = (V (GR ), E(GR )), where V (GR ) = R. For brevity, for a subset E 0 of E(G), we will use E 0 for E(GR ) to denote that E(GR ) = {{i, j} | ∃{u, v} ∈ E 0 , u ∈ Ri ∧ v ∈ Rj } ,

1 ≤ i, j ≤ k.

(21)

Since T ∗ is an alternating spanning tree, thus all vertices in V [G] are connected, the individual cycles Ri must be connected with each other as well, i.e., the graph GR = (R, T > ) is connected. We can choose an inclusion wise minimal T ⊥ ⊆ T > , such that the graph TR = (R, T ⊥ ) remains connected, i.e., TR is a spanning tree of GR , as in Fig. 2(a). We term the procedure for constructing alternating Hamiltonian cycles 2APX. Next, we give a brief summary of the construction procedure 2APX Step 1: Compute a minimum weight perfect matching M ∗ and a minimum weight alternating spanning tree T ∗ in G; Step 2: Let R := {Ri : i = 1, 2, . . . , k} be the cycle cover of G given by M∗ + TM; Step 3: Choose an inclusion wise minimal T ⊥ ⊆ T > such that TR = (R, T ⊥ ) is a spanning tree; Step 4: Construct a multigraph E2APX = (V (E2APX ), E(E2APX )), where V (E2APX ) = V (G), and E(E2APX ) = M ∗ ] T M ] 2 · T ⊥ (Fig. 2(b)); Step 5: Shortcut an Eulerian walk of E2APX to an alternating Hamiltonian cycle H2APX , preserving the edges from M ∗ .

8

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Rm Rj

:B :W

u

v q

Ri Rl

y

(a) (b) Fig. 2. (a) A representation of TR = (R, T ⊥ ). Nodes of R ( ), are individual cycles over V (G) = B∪W . (b) The resulting multigraph E2APX , arrows added to aid the image of traversing. The perfect matching M ∗ is given in bold gray lines, T M in slender black, and the two copies of T ⊥ in dashed lines.

The multigraph E2APX over the vertex set V (G) = B ∪ W in Fig. 2(b), has as its edge set a multiset sum of M ∗ , T M and two copies of T ⊥ . We need to show that this structure can be used to obtain a valid alternating cycle. As a first step, we will elaborate that there is an Eulerian walk. Lemma 2. The multigraph E2APX is Eulerian. Proof. We need to show that E2APX is connected, and every vertex has even degree w.r.t. E2APX . Connectedness follows from the fact that we sought the structure TR = (R, T ⊥ ) to be a spanning tree, where R is a cycle cover of the vertex set V (G) = B ∪ W . Every vertex in V (G) is of degree 2 w.r.t. the cycle cover R. Finally, we have added two copies of T ⊥ , hence the claim follows. t u Next we show how E2APX can be shortcut to give an alternating Hamiltonian cycle. Lemma 3. The Eulerian graph E2APX can always be shortcut to an alternating Hamiltonian cycle H2APX , preserving the edges from M ∗ . Proof. We will prove this claim by induction over the number of cycles k in the cycle cover R – Case k = 1: Trivial, this is H2APX ; – Case k > 1: Start from the observation that T ⊥ is bipartite. Therefore there must exist a certain q ∈ B connected to some y ∈ W by an arc {q, y} ∈ T ⊥ . Let q ∈ Ri and y ∈ Rj . Now, let u ∈ W (also u ∈ Ri ) such that {u, q} ∈ T M , and let v ∈ B (also v ∈ Rj ), such that {y, v} ∈ T M (Fig. 2(b)). We shortcut {{u, q}, {q, y}, {y, v}} by {u, v}, thus merging the two cycles Ri and Rj and decreasing the number of cycles by one. Note, all of the shortcut edges, {u, q}, {q, y} and {y, v} belong to T (either in T ⊥ ⊆ T > or T M ), thus edges in M ∗ are preserved intact. Lastly, due to the quadrangle inequality from Eq. (7), this shortcutting will not increase the total weight w(E2APX ). t u In the end, we will have obtained an alternating Hamiltonian cycle H2APX .

Approximating the Bipartite TSP and its Biased Generalization

4.2

9

Approximation Ratio

Next, we investigate the applicability of the 2APX procedure as an approximation algorithm. Lemma 4. For a given instance of the metric BBTSP, let H ∗ be an alternating −→ Hamiltonian cycle of minimal cost L(H ∗ ). Let the edge set H ∗ ⊂ E(H ∗ ) be ∗ traversed in the direction from B to W , so that the value L(H ) is parameterized −→ by some λ ∈ R+ as L(H ∗ ) = (β + λ)w(H ∗ ). For H2APX as the result from the 2APX procedure it holds L(H2APX ) ≤

β + 2λ + 1 L(H ∗ ). β+λ

(22)

Proof. In order to derive an upper bound of the cost L(H2APX ), we will retrace the steps from the construction process, and recall some of the bounds presented in Section 3, especially Subsection 3.2. First, recall that we chose a T ⊥ ⊆ T > , therefore w(T ⊥ ) ≤ w(T > ). It readily follows (see Eq. (18)) w(T ⊥ ) ≤ w(T ∗ ) − w(T M ). (23) → − ← − Let us partition E(H2APX ) into two disjoint matchings, H 2APX and H 2APX , → − ← − in such a way that H 2APX = M ∗ and H 2APX is a shortcut through T M ] T ⊥ ] T ⊥ , as in Lemma 3. We choose a traversal orientation such that exactly the edges → − of H 2APX are traversed in the direction from B to W . From the bias factor β of Eqs. (1), (11) and (13) → − ← − L(H2APX ) = βw( H 2APX ) + w( H 2APX ) ≤ βw(M ∗ ) + w(T M ) + 2w(T ⊥ ).

(24)

Recall the partition of a minimum cost alternating spanning tree from Eq. (18) and the related bounds from Eq. (23) and substitute them in Eq. (24). From this, and the fact that w(M ∗ ) ≤ w(T M ), we get L(H2APX ) ≤ βw(M ∗ ) + 2w(T ∗ ) − w(T M ) ≤ 2w(T ∗ ) + (β − 1)w(M ∗ ).

(25)

Next we substitute for M ∗ and T ∗ the bounds given with Eqs. (14) and (15) −→ ←− −→ L(H2APX ) ≤2(w(H ∗ ) + w(H ∗ )) + (β − 1)w(H ∗ ) −→ −→ =2(1 + λ)w(H ∗ ) + (β − 1)w(H ∗ ). Finally, following Eq. (20), the expression above leads to the claim.

(26) t u

10

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Lemma 4 gives the result announced in the Introduction, Eq. (2) 1+λ L(H2APX ) ≤1+ . ∗ L(H ) β+λ The result from Lemma 4 and the definition of an approximation ratio of Eq. (4) give the following result α2APX = 2, which holds true for any β ≥ 1 and λ ∈ R+ . However, Eq. (2) does provide us with insight of the behavior of L(H2APX ) for increasing values of β, and some reasonable finite upper bound on λ. 4.3

As a Part of a Composite Heuristic

As the previously known approximation ratio of 16/7 described in [17] relies on a composite heuristic, i.e., on a trade-off between two different procedures for building an alternating Hamiltonian path, we investigate a similar approach. For that purpose, we only briefly review a well known procedure for constructing an alternating Hamiltonian cycle in a given complete bipartite graph G = (B, W ; E), with |B| = |W | =: n. We term this procedure as procedure SWAP (the same procedure has been termed a matching based heuristic in [17].) The SWAP procedure has been described as a heuristic method for the swapping problem [2], and adopted to the bipartite TSP [3]. Briefly described, it is as follows Step 1: Find a minimum cost perfect matching M ∗ in G = (B, W ; E); Step 2: Build a ζ-approximate Hamiltonian cycle C 0 in G[B]; Step 3: Make an Eulerian multigraph ESWAP = (V (ESWAP ), E(ESWAP )), where V (ESWAP ) = V (G) and E(ESWAP ) = E(C 0 ) ] 2 · M ∗ ; Step 4: Appropriately shortcut an Eulerian walk in ESWAP to get an alternating Hamiltonian cycle HSWAP in G, preserving one copy of M ∗. The correctness and validity of the SWAP procedure is argued in more detail in, e.g., [2, 3, 17]. For the purpose of arriving to a suitable expression for a composite heuristic relying on the 2APX and SWAP procedures, we will present our bounds on L(HSWAP ). Analogous to Eq. (17), for a ζ-approximate C 0 of an optimal C ∗ we get −→ ←− w(C 0 ) ≤ ζw(C ∗ ) ≤ ζ w(H ∗ ) + w(H ∗ ) . (27) Since we can shortcut an Eulerian walk in ESWAP to obtain HSWAP in such a way that one copy of M ∗ is preserved, we can orient the traversal of HSWAP so that exactly the edges in M ∗ are traversed in the direction from B to W . Following Eqs. (14), (20) and (27) −→ L(HSWAP ) ≤ζ(1 + λ)w(H ∗ ) + (β + 1)w(M ∗ ) ≤

ζ(1 + λ) + β + 1 L(H ∗ ). β+λ

(28)

Approximating the Bipartite TSP and its Biased Generalization

11

Since from Lemma 1 we have that the extension of w over the edges of G[B] is symmetric and satisfies the triangle inequality, we can use, e.g., Christofides’ heuristic [6] to build a C 0 with ζ = 3/2. We propose a simple procedure which will compute both H2APX and HSWAP according to their respective construction procedures, and choose the one of lower cost. Let us term this procedure COM P and the resulting alternating Hamiltonian cycle HCOM P . From Eqs. (22) and (28) we get β + 2λ + 1 ∗ ζ(1 + λ) + β + 1 ∗ L(H ), L(H ) L(HCOM P ) ≤ min β+λ β+λ 2 ≤ 1+ L(H ∗ ). (29) ζ + β(2 − ζ) The trade-off in Eq. (3) is achieved for λ = to be called when ζ < 2. It readily follows αCOM P = 1 +

ζ 2−ζ ,

therefore, it only makes sense

2 , ζ + β(2 − ζ)

which is not dependent on a hidden instance-specific parameter, such as λ. 4.4

Computational Complexity

Without much deliberation we will state that all procedures undertaken to obtain an alternating Hamiltonian cycle have well known polynomial time implementations. An excellent source of information concerning the presented combinatorial structures as well as their algorithmic implementations can be found in [12, 16], as well as [8]. We will just state that the bottleneck procedure in the computation is finding a minimum cost alternating spanning tree T ∗ in the bipartite graph G = (B, W ; E) (|B| = |W | =: n), since it requires a call to a general matroid intersection algorithm, which in turn requires O(n7 ) time ([3, 8, 9, 11, 12, 16, 18]). As a consequence, we can state the following Theorem 1. The biased bipartite traveling salesman problem with a symmetric edge weight function satisfying the quadrangle inequality and a bias β ≥ 1 can be approximated within a constant factor α = 2, in polynomial time complexity.

5

Conclusion

We formalized the biased bipartite TSP (BBTSP) as a generalization of the symmetric bipartite TSP with quadrangle inequality by introducing a bias term β ≥ 1, which introduces asymmetry in the cost of an alternating Hamiltonian path. This generalization had been introduced as a means to better capture some features of industrial material handling scenarios. We presented a novel heuristic for building alternating Hamiltonian cycles in complete bipartite graphs. With that, obtained a first nontrivial approximation algorithm which improves the approximation factor of previously known

12

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

approaches to a constant 2, and showed that this approximation ratio holds for any value of the bias β ≥ 1. We also analyzed the performance of the proposed procedure for building alternating Hamiltonian cycles as a part of a composite heuristic, and derived an approximation ratio which benefits of both a better approximation for the metric TSP, and an increased value for the bias β. It is a standing question whether the constant bound 2 of the approximation ratio presented in this paper can be further improved by some algorithms similar to existing approaches for the standard metric TSP [7].

References 1. Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the Hamiltonian cycle problem for bipartite graphs. Journal of Information Processing 3(2) (1980) 73–76 2. Anily, S., Hassin, R.: The swapping problem. Networks 22(4) (1992) 419–433 3. Baltz, A., Srivastav, A.: Approximation algorithms for the Euclidean bipartite TSP. Operations Research Letters 33(4) (2005) 403 – 410 4. Chalasani, P., Motwani, R.: Approximating capacitated routing and delivery problems. SIAM Journal on Computing 28(6) (1999) 2133–2149 5. Chalasani, P., Motwani, R., Rao, A.: Algorithms for robot grasp and delivery. In: Proceedings of 2nd International Workshop on Algorithmic Foundations of Robotics. (1996) 347–362 6. Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976) 7. Cook, W.J.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press (2012) 8. Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications. OUP Oxford (2011) 9. Frank, A.: A weighed matroid intersection algorithm. Journal of Algorithms 2 (1981) 328–336 10. Frank, A., Triesch, E., Korte, B., Vygen, J.: On the bipartite traveling salesman problem. Technical Report 98866-OR, University of Bonn (1998) 11. Karuno, Y., Nagamochi, H., Shurbevski, A.: An approximation algorithm with factor two for a repetitive routing problem of grasp-and-delivery robots. Journal of Advanced Computational Intelligence and Intelligent Informatics 15(8) (2011) 1103–1108 12. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 3rd ed. Springer (2005) 13. Krishnamoorthy, M.: An NP-hard problem in bipartite graphs. SIGACT News 7 (1975) 26–26 14. Langston, M.A.: A study of composite heuristic algorithms. The Journal of the Operational Research Society 38(6) (1987) pp. 539–544 15. Sahni, S., Gonzalez, T.: P-complete approximation problems. Journal of the ACM 23(3) (1976) 555–565 16. Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer (2003)

Approximating the Bipartite TSP and its Biased Generalization

13

17. Shurbevski, A., Nagamochi, H., Karuno, Y.: Heuristics for a repetitive routing problem of a single grasp-and-delivery robot with an asymmetric edge cost function. In: 10th International Conference of the Society for Electronics, Telecommunications, Automatics and Informatics (ETAI 2011), CD-ROM Proceedings, A1-1. (2011) 18. Srivastav, A., Schroeter, H., Michel, C.: Approximation algorithms for pick-andplace robots. Annals of Operations Research 107(3) (2001) 321–338

2

Department of Applied Mathematics and Physics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan {shurbevski,nag}@amp.i.kyoto-u.ac.jp Department of Mechanical and System Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan [email protected]

Abstract. We examine a generalization of the symmetric bipartite traveling salesman problem (TSP) with quadrangle inequality, by extending the cost function of a Hamiltonian tour to include a bias factor β ≥ 1. The bias factor is known and given as a part of the input. We propose a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs, and show that it is an approximation algorithm for the gener1+λ alized problem with an approximation ratio of 1 + β+λ , where λ is a real parameter dependent on the problem instance. This expression is bounded above by a constant 2, for any positive real λ and β ≥ 1, which improves a previously reported approximation ratio of 16/7. As a part of a composite heuristic, the proposed procedure can contribute to an 2 , where ζ is an approximation ratio approximation ratio of 1 + ζ+β(2−ζ) for the metric TSP.

Keywords: combinatorial optimization; approximation algorithm; matroid intersection; material handling robot; bipartite TSP; biased cost

1

Introduction

The traveling salesman problem (TSP) is a landmark problem in combinatorial optimization (e.g., Cook [7]). Its bipartite analogue is as follows. Given a bipartite graph G = (B, W ; E) with an edge weight function w : E → R+ , find a shortest (w.r.t. w) alternating tour which visits every point of B ∪ W exactly once. We assume that the weight function w is symmetric and satisfies the quadrangle inequality (the bipartite analogue of the triangle inequality, see Eqs. (6) and (7)). We do so not only because do the above conditions suffice in many cases based on real world scenarios, but also because just like the TSP, it is hopeless ?

Full paper DOI:10.1007/978-3-319-04657-0_8, available on line at http://link. springer.com/chapter/10.1007%2F978-3-319-04657-0_8

2

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

to approximate the bipartite TSP within a constant factor in the general case, assuming that P 6= NP [10, 15]. The bipartite TSP has justly attracted attention due to its applicability in typical industrial settings where pick and place or grasp and delivery robots are employed with some material handling tasks [3–5, 10, 11, 18]. For the symmetric case, the best known approximation factor 2 has been independently reported by Chalasani et al. [5] and Frank et al. [10]. With a specific industrial scenario in mind, the bipartite TSP has been extended to account for additional transportation effort [17]. The motivation behind this generalization is to assign certain “difficulty” when transporting an item versus simply moving through space. This has been achieved by the means of a bias factor β ≥ 1. The bias factor extends the weight function w as follows ( βw(u, v), u ∈ B, v ∈ W, w(u, e v) = (1) w(u, v), u ∈ W, v ∈ B. To the best of our knowledge, Shurbevski et al. [17] gave the first account examining the presence of a bias factor, and at the same time, demonstrated a constant 16/7-factor approximation algorithm. The previously reported approximation ratio of 16/7 has been achieved by a composite heuristic (see, e.g., [14] for terminology relating to composite heuristics). In this paper, we present a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs and show that for the biased case it is an approximation algorithm with an approximation ratio of 1+λ (2) 1+ β+λ where λ is a real parameter which depends on the problem instance and cannot be known upfront. On one hand, the above expression is bounded by a constant 2 for any positive real λ and β ≥ 1, thus the proposed algorithm has a constant factor approximation ratio, improving the one from [17]. On the other hand, for a finite λ, the above expression approaches 1 as β grows larger. The presented approach by itself does not rely on approximating the metric TSP, however it can be used as a part of a composite heuristic to achieve an approximation ratio of 2 1+ , (3) ζ + β(2 − ζ) where 1 < ζ ≤ 2 is an approximation ratio for the metric TSP. The expression from Eq. (3) is also bounded above by a constant 2, but it is not dependent on an instance-specific parameter, and has a clear relationship with the bias β for a fixed ζ < 2.

2

Preliminaries

The set of reals (resp., nonnegative reals) is denoted by R (resp., R+ ).

Approximating the Bipartite TSP and its Biased Generalization

3

In general, for a minimization problem P, let P ∗ be the value of an optimal solution. An approximation algorithm ALG is such that for any instance of P, it can produce a feasible solution of value P 0 . We call the value 0 P (4) αALG = sup P∗ the approximation factor of algorithm ALG, and usually say that ALG is an αALG -approximation algorithm. We use standard notation from graph theory; the ordered pair G = (V, E) is a connected undirected graph. The vertex set and the edge set of G are denoted by V (G) and E(G), respectively. We allow for parallel edges, or think of G = (V, E) as a multigraph. Thus, E(G) is a multiset of elements in V ×V . (We will make use of the multiset Uk sum function, denoted by the symbol ], as well as the shorthand k · E for i=1 E.) We use {u, v}, u, v ∈ V (G) to reference any and all e ∈ E(G) such that e is incident with u and v. For u ∈ V (G), dG (u) denotes the degree of the node u in the graph G. A graph is weighted if we are given some weight function w : E(G) of edges E 0 ⊆ E, P → R+ over the graph’s edges. For any subset 0 0 w(E for a subgraph G of G, w(G0 ) denotes P ) denotes e∈E 0 w(e). Similarly, 0 0 e∈E(G0 ) w(e). A subgraph G of G is spanning if V (G ) = V (G). We assume that all parallel edges are of the same weight, and ∀e ∈ E(G), e = {u, v}, we equate the expressions w(e) and w(u, v). The weight function w is said to be symmetric if w(u, v) = w(v, u), ∀e = {u, v} ∈ E(G), (5) and that it satisfies the triangle inequality if w(u, v) ≤ w(u, q) + w(q, v),

∀q, u, v ∈ V (G).

(6)

A complete bipartite graph G = (B, W ; E) is such that V (G) = B ∪ W , B ∩ W = ∅, and E(G) = B × W . A property similar to the triangle inequality can be extended over complete bipartite graphs, into the quadrangle inequality w(u, v) ≤ w(u, q) + w(q, y) + w(y, v),

∀u, y ∈ B, q, v ∈ W.

(7)

For a complete graph induced by a set of vertices B, we write G[B]. By definition, V (G[B]) = B and E(G[B]) = B × B. Let G = (B, W ; E) be a given bipartite graph with an edge weight function w : E(G) → R+ , and G[B] is exactly the complete graph induced by the partition B. Often in practice the vertex sets are in fact points from some metric space and the distance in this space serves as an edge weight function. In such a case, the edge weight function of G[B] is defined by the distance function in the metric space. However, if we are only given a bipartite graph G = (B, W ; E) with an edge weight function w : E(G) → R+ , we can extend the edge weight function over the induced graph G[B] as such w(u, y) = min {w(u, q) + w(q, y)} q∈W

∀ u, y ∈ B.

(8)

4

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Lemma 1. For a given complete bipartite graph G(B, W ; E) with a symmetric edge weight function w : E(G) → R+ satisfying the quadrangle inequality, let G[B] be the complete graph induced by the vertex partition B. The extension of w as an edge weight function of G[B] of Eq. (8) is symmetric and satisfies the triangle inequality. Given a graph G = (V, E), a Hamiltonian cycle H is a connected spanning subgraph of G such that dH (u) = 2,

∀u ∈ V (G).

(9)

The problem of finding a Hamiltonian cycle H of minimum w(H) is commonly referred to as the traveling salesman problem (TSP). For a complete bipartite graph G = (B, W ; E), with |B| = |W |, let n := |B|(= |W |) and let σ and τ be permutations on the points of B and W , respectively. A traversal of a Hamiltonian cycle H in G is of the form σ(1) → τ (1) → σ(2) → · · · → τ (n − 1) → σ(n) → τ (n) → σ(1).

(10)

We term Hamiltonian cycles in bipartite graphs alternating, for points in B and W appear alternately. When using an indexing device i = 1, . . . , n, we allow it to wrap around, i.e. ( i + n, i ≤ 0, i := i − n, i > n. As subgraphs of G, Hamiltonian cycles are undirected. However, once we settle for a way to traverse them, they assume an orientation. In addition to the edge weight w, we are concerned with a bias factor β ≥ 1. The bias factor impacts bipartite graphs as in Eq. (1). Assuming a traversal orientation as in Eq. (10), we introduce the biased cost L for alternating cycles L(H) = β

n X i=1

w(σ(i), τ (i)) +

n X

w(τ (i), σ(i + 1)).

(11)

i=1

We are now prepared to state the bipartite analogue of the metric TSP in face of the bias factor β ≥ 1. The biased bipartite traveling salesman problem – BBTSP Instance: A complete bipartite graph G = (B, W ; E), with |B| = |W |, a symmetric weight function w : E(G) → R+ which satisfies the quadrangle inequality, and a bias factor β ≥ 1. Task: Find an alternating Hamiltonian cycle H ∗ in G such that L(H ∗ ) is minimized. In this paper we focus exclusively on the version of the BBTSP where the edge weight function w is symmetric and satisfies the quadrangle inequality. We settle for this limitation because it has been shown [1, 10, 13, 15] that the bipartite TSP is not only NP-hard to solve, but also that in the general case, there is no constant factor approximation under the assumption that P 6= NP.

Approximating the Bipartite TSP and its Biased Generalization

3

5

Building Blocks

In this section we will exhibit some of the known lower bounds on the value of an optimal solution for the BBTSP, as well as add a few new insights into their correlations. The presented lower bounds are structures well known in combinatorial optimization, and will serve as building blocks for a new procedure for constructing alternating Hamiltonian cycles in bipartite graphs. 3.1

Known Lower Bounds of the BBTSP

We present some of the observations made in [17] concerning the lower bounds of an optimal solution for the BBTSP. Our analysis mainly concerns two combinatorial structures in bipartite graphs; perfect matchings, and alternating spanning trees. We will just briefly state their definitions. Let G = (B, W ; E) be a (weighted) complete bipartite graph with an edge weight function w : E(G) → R+ and |B| = |W | =: n. The edge weight function w is assumed symmetric and satisfying the quadrangle inequality (Eq. (7)). A perfect matching M ⊂ E(G) is such that there is exactly one edge in M incident with any u ∈ V (G). An alternating spanning tree T (illustrated in Fig. 1(a)) is a connected acyclic spanning subgraph of G such that dT (u) ≤ 2, ∀u ∈ B. (12) Both perfect matchings and alternating spanning trees are well studied combinatorial structures, e.g., [12, 16], and there exist polynomial time algorithms for computing perfect matchings and alternating spanning trees (of minimum weight) in bipartite graphs. Henceforth, let M ∗ denote a perfect matching in G of minimum weight w(M ∗ ), and T ∗ an alternating spanning tree with minimum w(T ∗ ). Given an instance of the BBTSP, let H ∗ be an optimal solution, which minimizes the biased cost L(H ∗ ). The edges of E(H ∗ ) can be decomposed into two −→ ←− disjoint perfect matchings, H ∗ and H ∗ , as in Fig. 1(b). Without loss of general−→ ity, we assume H ∗ is to be traversed as indicated by arrows in Fig. 1(b), and H ∗ solely accounts for the bias term. The biased path cost L(H ∗ ) is given by

It surely holds

−→ ←− L(H ∗ ) = βw(H ∗ ) + w(H ∗ ).

(13)

−→ ←− w(M ∗ ) ≤ w(H ∗ ) ≤ w(H ∗ ).

(14)

Concerning alternating spanning trees in G, note that w(T ∗ ) is a lower bound of the weight of an alternating Hamiltonian cycle disregarding the bias factor, i.e., −→ ←− w(T ∗ ) ≤ w(H ∗ ) + w(H ∗ ).

(15)

6

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

v :B :W

→

H*

Ĉ

←* H

(a) (b) Fig. 1. (a) An alternating spanning tree T . (b) A minimum cost alternating Hamil−→ ←− tonian path H ∗ of G. The subsets of edges H ∗ (bold gray arrows) and H ∗ (slender ˆ on G[B] is given in black arrows) form two disjoint perfect matchings. The shortcut C dashed lines.

Observing the graph G[B] induced by the vertex partition B, we can see that an alternating Hamiltonian cycle in G does in fact visit each vertex in B exactly once, and can be shortcut to a Hamiltonian cycle of G[B]. We will use the extended w from Eq. (8) for G[B]. For an optimal alternating Hamiltonian cycle H ∗ , let Cˆ be the resulting shortcut, as given in Fig. 1(b). Due to Eq. (8) we have → ←− ˆ ≤ w(− w(C) H ∗ ) + w(H ∗ ). (16) Consequently, for an optimal (w.r.t. the extended w) Hamiltonian cycle C ∗ in G[B] it holds → ←− ˆ ≤ w(− w(C ∗ ) ≤ w(C) H ∗ ) + w(H ∗ ). (17) 3.2

Further Observations

We would like to bring a special attention to an observation with respect to the structures presented above, alternating spanning trees and perfect matchings. Let M ∗ and T ∗ be a minimum weight perfect matching and a minimum weight alternating spanning tree in a given bipartite graph G, respectively. Owing to its special structure any alternating spanning tree in G contains a perfect matching. Therefore, let T M ⊂ E(T ∗ ) denote the edge set forming a perfect matching, and T > the remaining edges of the alternating tree, i.e. T > = E(T ∗ )\T M . It simply holds w(T ∗ ) = w(T M ) + w(T > ). (18) We present our view of the structure of an optimal solution, H ∗ , with L(H ∗ ) = −→ ←− βw(H ∗ ) + w(H ∗ ), (see Eq. (13)). We introduce a parameter λ ∈ R+ as ←− w(H ∗ ) λ= (19) −→ . w(H ∗ ) Then, for the cost of an optimal tour H ∗ we can write −→ L(H ∗ ) = (β + λ)w(H ∗ ).

(20)

Approximating the Bipartite TSP and its Biased Generalization

7

For a given instance of the BBTSP, the value of the parameter λ cannot be known without solving it exactly. However, for the purpose of our exposition, it suffices that λ ∈ R+ .

4

A New Approximation Algorithm

In this section we present a procedure for building an alternating Hamiltonian cycle in a given bipartite graph G = (B, W ; E) with |B| = |W |. We show that if the graph G is endowed with a positive symmetric edge weight function w which satisfies the quadrangle inequality, this procedure can be used as an approximation algorithm for the BBTSP. The procedure for building an alternating Hamiltonian cycle does not rely on approximating the metric TSP. 4.1

Construction

Let G = (B, W ; E), be a bipartite graph with |B| = |W | =: n. Let w : E(G) → R+ be a symmetric edge weight function satisfying the quadrangle inequality. Let M ∗ and T ∗ be a perfect matching and an alternating spanning tree in G of minimum w(M ∗ ) and w(T ∗ ), respectively. We bring to attention the union of M ∗ and T ∗ . As observed in Section 3.1, the alternating tree T ∗ contains a perfect matching, T M . The union of T M and M ∗ forms a cycle cover of G. Let there be k ≤ n individual cycles, which we will denote by R := {Ri : i = 1, 2, . . . , k}. We can think of elements of R as nodes, and define a graph GR = (V (GR ), E(GR )), where V (GR ) = R. For brevity, for a subset E 0 of E(G), we will use E 0 for E(GR ) to denote that E(GR ) = {{i, j} | ∃{u, v} ∈ E 0 , u ∈ Ri ∧ v ∈ Rj } ,

1 ≤ i, j ≤ k.

(21)

Since T ∗ is an alternating spanning tree, thus all vertices in V [G] are connected, the individual cycles Ri must be connected with each other as well, i.e., the graph GR = (R, T > ) is connected. We can choose an inclusion wise minimal T ⊥ ⊆ T > , such that the graph TR = (R, T ⊥ ) remains connected, i.e., TR is a spanning tree of GR , as in Fig. 2(a). We term the procedure for constructing alternating Hamiltonian cycles 2APX. Next, we give a brief summary of the construction procedure 2APX Step 1: Compute a minimum weight perfect matching M ∗ and a minimum weight alternating spanning tree T ∗ in G; Step 2: Let R := {Ri : i = 1, 2, . . . , k} be the cycle cover of G given by M∗ + TM; Step 3: Choose an inclusion wise minimal T ⊥ ⊆ T > such that TR = (R, T ⊥ ) is a spanning tree; Step 4: Construct a multigraph E2APX = (V (E2APX ), E(E2APX )), where V (E2APX ) = V (G), and E(E2APX ) = M ∗ ] T M ] 2 · T ⊥ (Fig. 2(b)); Step 5: Shortcut an Eulerian walk of E2APX to an alternating Hamiltonian cycle H2APX , preserving the edges from M ∗ .

8

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Rm Rj

:B :W

u

v q

Ri Rl

y

(a) (b) Fig. 2. (a) A representation of TR = (R, T ⊥ ). Nodes of R ( ), are individual cycles over V (G) = B∪W . (b) The resulting multigraph E2APX , arrows added to aid the image of traversing. The perfect matching M ∗ is given in bold gray lines, T M in slender black, and the two copies of T ⊥ in dashed lines.

The multigraph E2APX over the vertex set V (G) = B ∪ W in Fig. 2(b), has as its edge set a multiset sum of M ∗ , T M and two copies of T ⊥ . We need to show that this structure can be used to obtain a valid alternating cycle. As a first step, we will elaborate that there is an Eulerian walk. Lemma 2. The multigraph E2APX is Eulerian. Proof. We need to show that E2APX is connected, and every vertex has even degree w.r.t. E2APX . Connectedness follows from the fact that we sought the structure TR = (R, T ⊥ ) to be a spanning tree, where R is a cycle cover of the vertex set V (G) = B ∪ W . Every vertex in V (G) is of degree 2 w.r.t. the cycle cover R. Finally, we have added two copies of T ⊥ , hence the claim follows. t u Next we show how E2APX can be shortcut to give an alternating Hamiltonian cycle. Lemma 3. The Eulerian graph E2APX can always be shortcut to an alternating Hamiltonian cycle H2APX , preserving the edges from M ∗ . Proof. We will prove this claim by induction over the number of cycles k in the cycle cover R – Case k = 1: Trivial, this is H2APX ; – Case k > 1: Start from the observation that T ⊥ is bipartite. Therefore there must exist a certain q ∈ B connected to some y ∈ W by an arc {q, y} ∈ T ⊥ . Let q ∈ Ri and y ∈ Rj . Now, let u ∈ W (also u ∈ Ri ) such that {u, q} ∈ T M , and let v ∈ B (also v ∈ Rj ), such that {y, v} ∈ T M (Fig. 2(b)). We shortcut {{u, q}, {q, y}, {y, v}} by {u, v}, thus merging the two cycles Ri and Rj and decreasing the number of cycles by one. Note, all of the shortcut edges, {u, q}, {q, y} and {y, v} belong to T (either in T ⊥ ⊆ T > or T M ), thus edges in M ∗ are preserved intact. Lastly, due to the quadrangle inequality from Eq. (7), this shortcutting will not increase the total weight w(E2APX ). t u In the end, we will have obtained an alternating Hamiltonian cycle H2APX .

Approximating the Bipartite TSP and its Biased Generalization

4.2

9

Approximation Ratio

Next, we investigate the applicability of the 2APX procedure as an approximation algorithm. Lemma 4. For a given instance of the metric BBTSP, let H ∗ be an alternating −→ Hamiltonian cycle of minimal cost L(H ∗ ). Let the edge set H ∗ ⊂ E(H ∗ ) be ∗ traversed in the direction from B to W , so that the value L(H ) is parameterized −→ by some λ ∈ R+ as L(H ∗ ) = (β + λ)w(H ∗ ). For H2APX as the result from the 2APX procedure it holds L(H2APX ) ≤

β + 2λ + 1 L(H ∗ ). β+λ

(22)

Proof. In order to derive an upper bound of the cost L(H2APX ), we will retrace the steps from the construction process, and recall some of the bounds presented in Section 3, especially Subsection 3.2. First, recall that we chose a T ⊥ ⊆ T > , therefore w(T ⊥ ) ≤ w(T > ). It readily follows (see Eq. (18)) w(T ⊥ ) ≤ w(T ∗ ) − w(T M ). (23) → − ← − Let us partition E(H2APX ) into two disjoint matchings, H 2APX and H 2APX , → − ← − in such a way that H 2APX = M ∗ and H 2APX is a shortcut through T M ] T ⊥ ] T ⊥ , as in Lemma 3. We choose a traversal orientation such that exactly the edges → − of H 2APX are traversed in the direction from B to W . From the bias factor β of Eqs. (1), (11) and (13) → − ← − L(H2APX ) = βw( H 2APX ) + w( H 2APX ) ≤ βw(M ∗ ) + w(T M ) + 2w(T ⊥ ).

(24)

Recall the partition of a minimum cost alternating spanning tree from Eq. (18) and the related bounds from Eq. (23) and substitute them in Eq. (24). From this, and the fact that w(M ∗ ) ≤ w(T M ), we get L(H2APX ) ≤ βw(M ∗ ) + 2w(T ∗ ) − w(T M ) ≤ 2w(T ∗ ) + (β − 1)w(M ∗ ).

(25)

Next we substitute for M ∗ and T ∗ the bounds given with Eqs. (14) and (15) −→ ←− −→ L(H2APX ) ≤2(w(H ∗ ) + w(H ∗ )) + (β − 1)w(H ∗ ) −→ −→ =2(1 + λ)w(H ∗ ) + (β − 1)w(H ∗ ). Finally, following Eq. (20), the expression above leads to the claim.

(26) t u

10

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

Lemma 4 gives the result announced in the Introduction, Eq. (2) 1+λ L(H2APX ) ≤1+ . ∗ L(H ) β+λ The result from Lemma 4 and the definition of an approximation ratio of Eq. (4) give the following result α2APX = 2, which holds true for any β ≥ 1 and λ ∈ R+ . However, Eq. (2) does provide us with insight of the behavior of L(H2APX ) for increasing values of β, and some reasonable finite upper bound on λ. 4.3

As a Part of a Composite Heuristic

As the previously known approximation ratio of 16/7 described in [17] relies on a composite heuristic, i.e., on a trade-off between two different procedures for building an alternating Hamiltonian path, we investigate a similar approach. For that purpose, we only briefly review a well known procedure for constructing an alternating Hamiltonian cycle in a given complete bipartite graph G = (B, W ; E), with |B| = |W | =: n. We term this procedure as procedure SWAP (the same procedure has been termed a matching based heuristic in [17].) The SWAP procedure has been described as a heuristic method for the swapping problem [2], and adopted to the bipartite TSP [3]. Briefly described, it is as follows Step 1: Find a minimum cost perfect matching M ∗ in G = (B, W ; E); Step 2: Build a ζ-approximate Hamiltonian cycle C 0 in G[B]; Step 3: Make an Eulerian multigraph ESWAP = (V (ESWAP ), E(ESWAP )), where V (ESWAP ) = V (G) and E(ESWAP ) = E(C 0 ) ] 2 · M ∗ ; Step 4: Appropriately shortcut an Eulerian walk in ESWAP to get an alternating Hamiltonian cycle HSWAP in G, preserving one copy of M ∗. The correctness and validity of the SWAP procedure is argued in more detail in, e.g., [2, 3, 17]. For the purpose of arriving to a suitable expression for a composite heuristic relying on the 2APX and SWAP procedures, we will present our bounds on L(HSWAP ). Analogous to Eq. (17), for a ζ-approximate C 0 of an optimal C ∗ we get −→ ←− w(C 0 ) ≤ ζw(C ∗ ) ≤ ζ w(H ∗ ) + w(H ∗ ) . (27) Since we can shortcut an Eulerian walk in ESWAP to obtain HSWAP in such a way that one copy of M ∗ is preserved, we can orient the traversal of HSWAP so that exactly the edges in M ∗ are traversed in the direction from B to W . Following Eqs. (14), (20) and (27) −→ L(HSWAP ) ≤ζ(1 + λ)w(H ∗ ) + (β + 1)w(M ∗ ) ≤

ζ(1 + λ) + β + 1 L(H ∗ ). β+λ

(28)

Approximating the Bipartite TSP and its Biased Generalization

11

Since from Lemma 1 we have that the extension of w over the edges of G[B] is symmetric and satisfies the triangle inequality, we can use, e.g., Christofides’ heuristic [6] to build a C 0 with ζ = 3/2. We propose a simple procedure which will compute both H2APX and HSWAP according to their respective construction procedures, and choose the one of lower cost. Let us term this procedure COM P and the resulting alternating Hamiltonian cycle HCOM P . From Eqs. (22) and (28) we get β + 2λ + 1 ∗ ζ(1 + λ) + β + 1 ∗ L(H ), L(H ) L(HCOM P ) ≤ min β+λ β+λ 2 ≤ 1+ L(H ∗ ). (29) ζ + β(2 − ζ) The trade-off in Eq. (3) is achieved for λ = to be called when ζ < 2. It readily follows αCOM P = 1 +

ζ 2−ζ ,

therefore, it only makes sense

2 , ζ + β(2 − ζ)

which is not dependent on a hidden instance-specific parameter, such as λ. 4.4

Computational Complexity

Without much deliberation we will state that all procedures undertaken to obtain an alternating Hamiltonian cycle have well known polynomial time implementations. An excellent source of information concerning the presented combinatorial structures as well as their algorithmic implementations can be found in [12, 16], as well as [8]. We will just state that the bottleneck procedure in the computation is finding a minimum cost alternating spanning tree T ∗ in the bipartite graph G = (B, W ; E) (|B| = |W | =: n), since it requires a call to a general matroid intersection algorithm, which in turn requires O(n7 ) time ([3, 8, 9, 11, 12, 16, 18]). As a consequence, we can state the following Theorem 1. The biased bipartite traveling salesman problem with a symmetric edge weight function satisfying the quadrangle inequality and a bias β ≥ 1 can be approximated within a constant factor α = 2, in polynomial time complexity.

5

Conclusion

We formalized the biased bipartite TSP (BBTSP) as a generalization of the symmetric bipartite TSP with quadrangle inequality by introducing a bias term β ≥ 1, which introduces asymmetry in the cost of an alternating Hamiltonian path. This generalization had been introduced as a means to better capture some features of industrial material handling scenarios. We presented a novel heuristic for building alternating Hamiltonian cycles in complete bipartite graphs. With that, obtained a first nontrivial approximation algorithm which improves the approximation factor of previously known

12

Aleksandar Shurbevski, Hiroshi Nagamochi, and Yoshiyuki Karuno

approaches to a constant 2, and showed that this approximation ratio holds for any value of the bias β ≥ 1. We also analyzed the performance of the proposed procedure for building alternating Hamiltonian cycles as a part of a composite heuristic, and derived an approximation ratio which benefits of both a better approximation for the metric TSP, and an increased value for the bias β. It is a standing question whether the constant bound 2 of the approximation ratio presented in this paper can be further improved by some algorithms similar to existing approaches for the standard metric TSP [7].

References 1. Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the Hamiltonian cycle problem for bipartite graphs. Journal of Information Processing 3(2) (1980) 73–76 2. Anily, S., Hassin, R.: The swapping problem. Networks 22(4) (1992) 419–433 3. Baltz, A., Srivastav, A.: Approximation algorithms for the Euclidean bipartite TSP. Operations Research Letters 33(4) (2005) 403 – 410 4. Chalasani, P., Motwani, R.: Approximating capacitated routing and delivery problems. SIAM Journal on Computing 28(6) (1999) 2133–2149 5. Chalasani, P., Motwani, R., Rao, A.: Algorithms for robot grasp and delivery. In: Proceedings of 2nd International Workshop on Algorithmic Foundations of Robotics. (1996) 347–362 6. Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976) 7. Cook, W.J.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press (2012) 8. Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications. OUP Oxford (2011) 9. Frank, A.: A weighed matroid intersection algorithm. Journal of Algorithms 2 (1981) 328–336 10. Frank, A., Triesch, E., Korte, B., Vygen, J.: On the bipartite traveling salesman problem. Technical Report 98866-OR, University of Bonn (1998) 11. Karuno, Y., Nagamochi, H., Shurbevski, A.: An approximation algorithm with factor two for a repetitive routing problem of grasp-and-delivery robots. Journal of Advanced Computational Intelligence and Intelligent Informatics 15(8) (2011) 1103–1108 12. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 3rd ed. Springer (2005) 13. Krishnamoorthy, M.: An NP-hard problem in bipartite graphs. SIGACT News 7 (1975) 26–26 14. Langston, M.A.: A study of composite heuristic algorithms. The Journal of the Operational Research Society 38(6) (1987) pp. 539–544 15. Sahni, S., Gonzalez, T.: P-complete approximation problems. Journal of the ACM 23(3) (1976) 555–565 16. Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer (2003)

Approximating the Bipartite TSP and its Biased Generalization

13

17. Shurbevski, A., Nagamochi, H., Karuno, Y.: Heuristics for a repetitive routing problem of a single grasp-and-delivery robot with an asymmetric edge cost function. In: 10th International Conference of the Society for Electronics, Telecommunications, Automatics and Informatics (ETAI 2011), CD-ROM Proceedings, A1-1. (2011) 18. Srivastav, A., Schroeter, H., Michel, C.: Approximation algorithms for pick-andplace robots. Annals of Operations Research 107(3) (2001) 321–338